The next stage is the input gate, which consists of both a Sigmoid function (\( \sigma_i \)), which decide what percentage of the input will be stored in the long-term memory, and the \( \tanh_i \) function, which decide what is the full memory that can be stored in the long term memory. When these results are calculated and multiplied together, it is added to the cell state or stored in the long-term memory, denoted as \( \oplus \).
We have
$$ \mathbf{i}^{(t)} = \sigma_g(W_i\mathbf{x}^{(t)} + U_i\mathbf{h}^{(t-1)} + \mathbf{b}_i), $$and
$$ \mathbf{\tilde{c}}^{(t)} = \tanh(W_c\mathbf{x}^{(t)} + U_c\mathbf{h}^{(t-1)} + \mathbf{b}_c), $$again the \( W \) and \( U \) are the weights.